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In this paper, we consider the problem of task-space tracking control of robot manipulators. Based on a quaternion representation of the end-effector orientation, we design a class of task-space controllers that ensure asymptotic end-effector position and orientation tracking. To facilitate the control design, we first develop model-based and adaptive full-state feedback controllers. We then present a model-based output feedback controller that eliminates link velocity measurements via a model-based observer. The application of the proposed control strategy to redundant robots is also discussed. Simulation results based on a six-link manipulator system are presented for the output feedback controller.